Every commutative ring has a geometry — Spec(Z), Spec(k[x]), and the points you never knew you were missing
In classical algebraic geometry, the points of a variety correspond to maximal ideals of a coordinate ring. Grothendieck's key insight was to enlarge this: instead of using only maximal ideals, we use all prime ideals. The resulting space, Spec(R), is called the spectrum of the ring R. This seemingly small change — including primes that aren't maximal — unlocks an enormous amount of geometric structure and makes algebraic geometry work over any commutative ring, not just polynomial rings over a field.
The simplest and most illuminating example is Spec(ℤ), the spectrum of the integers. Its points are the prime ideals (2), (3), (5), (7), … — one for each prime number — plus a mysterious generic point (0). The generic point belongs to every nonempty open set and "spreads out" over the entire space. If you've worked through the Algebraic Geometry module, think of Spec as the upgrade that lets geometry talk to number theory.
One closed point per prime, plus the generic point (0). Spec(Z) is the "arithmetic line."
A point of Spec(R) is a prime ideal 𝔭 ⊆ R. When R = k[x1, …, xn]/I is the coordinate ring of a variety, the maximal ideals of R correspond to the classical points of the variety (over an algebraically closed field). But Spec(R) also contains non-maximal primes — these correspond to irreducible subvarieties. For instance, in Spec(k[x, y]), the prime (y) is a point representing the entire x-axis, not a single location.
The closed points of Spec(R) are precisely the maximal ideals. These are the "ordinary" points that behave like classical variety points. Every other prime ideal corresponds to a "thicker" piece of geometry — a generic point of an irreducible closed subset. In Spec(ℤ), the closed points are (2), (3), (5), … while the generic point (0) sits underneath them all, belonging to every open set.
In the classical setting, you start with an algebraically closed field k, form polynomial rings, and study zero sets. The Nullstellensatz guarantees a bijection between points of a variety and maximal ideals of its coordinate ring. Spec generalizes this in two directions: it works over any commutative ring (not just k-algebras), and it sees all prime ideals (not just maximal ones).
This generalization is essential for arithmetic geometry (studying varieties over ℤ or ℤp) and for seeing families of varieties as single geometric objects. A morphism Spec(ℤ[x]/(x² + 1)) → Spec(ℤ) encodes the arithmetic of the Gaussian integers over all primes simultaneously — something impossible in the classical framework.